Based on non-rigorous but sophisticated considerations, physicists have put forward predictions on the probabilistic nature of classical problem in combinatorics such as the random graph coloring problem. The key object of interest from the physics point of view is the Boltzmann distribution. In the case of random graph coloring, this is just the uniform distribution on the set of all $k$-colorings of the graph. In this talk I will investigate the spatial mixing properties of the Boltzmann distribution of the random graph $G(n,d/n)$.

A lattice in $\mathbb R^n$ is called admissible if there exists a positive constant $c$ such that for every non-zero lattice point
the modulus of the product
of the coordinates is at least $c$. Generalising results of Hardy and Littlewood for $n=2$
Skriganov has shown that the lattice points of an admissible
lattice are extremely uniform distributed in aligned boxes. However, the set of admissible lattices in $SL_n(\mathbb R)/SL_n(\mathbb Z)$ is a
null set. We generalise the notion of admissibility and introduce a quantity to measure its failure. We then
shall discuss recent discrepancy results for ``weakly admissible'' lattices.

In a recent work Masser and Zannier showed that there are at most
finitely many complex numbers $\lambda \neq 0,1$ such that the points
$(2, \sqrt{2(2-\lambda)})$ and $(3, \sqrt{6(3-\lambda)})$ are
simultaneously torsion on the Legendre elliptic curve $E_\lambda$ of
equation $y^2=x(x-1)(x-\lambda)$. This is a special case of
conjectures about Unlikely Intersections on families of abelian
varieties, proved later in the two dimensional case by the same
authors. As a natural higher dimensional extension, we considered the
case of three points $(2, \sqrt{2(2-\lambda)})$, $(3,
\sqrt{6(3-\lambda)})$ and $(5, \sqrt{20(5-\lambda)})$ and proved that
there are at most finitely many $\lambda \neq 0,1$ such that these
three points satisfy two independent linear relations on $E_\lambda$.
This is a special case of a more general result in the framework of
the conjectures mentioned above.
This is joint work with L. Capuano.

Abstract:
Let us consider the 2-parameter family of real quadratic fields with square-free discriminant $D=(AN)^2+4A$ for positive odd integers $A$ and $N$. The class number one problem for this family is analogous to the class number one problem for imaginary quadratic fields but up to now the only known solution used the generalized Riemann hypothesis. Still there were unconditional results about certain subfamilies. In this talk the author would show how combining methods of Biró, a result from her PhD thesis and computer calculations we solve the class number one problem for this family. This is a joint work with A. Biró.

Strukturtheorie-Seminar

The discrete groups acting nicely on the plane, called planar discontinous groups, are a classical topic, closely related to e.g. surface groups. Planar discontinous groups have Cayley graphs that can be embedded in the plane without accumulation points of vertices, like for example the familiar tessellations of the hyperbolic plain, or Escher's tilings. There are however planar Cayley graphs all embeddings of which must have accumulation points of vertices. These graphs, and their groups, were far less well understood, and an open question due to Droms et. al. was whether they can be effectively enumerated.

We settle this question in the affirmative. The main part of our proof is to show that such groups admit nice presentations where the planarity 'can be seen' in an automatic way.

Bounds for the return probability of random walks on invariant percolations by random discs on amenable transitive graphs are derived which enable estimating the number of connected components per vertex. The scheme allows for a robustness against `noise' which the traditional union find algorithms (such as the Hoshen-Kopelman method) don't have.

Spectral asymptotics of second order differential operators of the form d/dm d/dx on the real line are well known if m is a self similar measure with compact support. We extend the results to some more general cases such as random fractal measures. Moreover, we give a representation of the eigenfunctions as generalized trigonometric functions.

The results were obtained in collaboration with Peter Arzt.

Strukturtheorie-Seminar

We consider a random walk obtained by subordinating the simple random walk on the d-dimensional integer lattice according to the procedure of discrete subordination introduced by A. Bendikov and L. Saloff-Coste. The corresponding Laplace exponent is a Bernstein function $\psi$ which is assumed to vary regularly at zero of index $0<\alpha <2$.

We investigate the asymptotic behaviour of the potential kernel of the associated discrete subordinator in two cases:

1. For $1<\alpha <2$ we use the celebrated renewal theorem of Garsia and
Lamperti.

2. For $0<\alpha <2$ we additionally assume that $\psi$ is a
special Bernstein function.

As an application of the asymptotics of the potential kernel we obtain the asymptotic behaviour (at infinity) of the Green function of the subordinated random walk.

Joint work with A. Bendikov

Discrete Mathematics Seminar

A bar-and-joint framework in the plane with degree of freedom 1 is called a mechanism. It is well-known that the operations of 0-extension and 1-extension, the so called Henneberg moves, can always be performed on a framework so that its degree of freedom is preserved. It was conjectured that for a mechanism in generic position these operations can be performed without restricting its motion. We provide a counterexample (This is joint work with B. Jackson, T. Jordan, H. Servatius)

be given, where $\varphi(z)=kz + d_2 z^2+ \ldots$ with $|k| >1$ and $a(z)=a_0
+ a_1 z + \ldots, a_0 \neq 0$, are given local analytic functions.
Then we show that all formal solutions $f, f(z)=c_1z+ c_2 z^2+ \ldots$
with $f(0)=0$ and $f$ not zero everywhere are local analytic.
Furthermore we are interested in normal forms of this equation.

Markov point processes are a spatial generalisation of the memoryless property of Markov chains. They play a key role in statistical mechanics and spatial probability theory. A Markov random field is specified by a consistent family of conditional distributions on finite volume regions with boundary conditions. In the infinite volume limit, such a specification gives rise to one or more Gibbs measures. The existence of only a unique or several Gibbs measures lies
behind the phase transitions in statistical mechanics. Disagreement percolation for lattice models by Maes and van den Berg is a technique to compare the competing influence of different boundary conditions with a product field. In other words, it relates the question of uniqueness of the Gibbs measure to percolation type problems. We generalise the technique to point processes and demonstrate the best possible improvement in the case of the hard-sphere model.

The Laplace-Beltrami operator on a Riemannian manifold can be defined by variation of the Dirichlet energy functional. This fact is basic to the generalization of the Laplacian to discrete surfaces and leads to the well established cotangent formula on triangle meshes. We will use this variational approach to derive a Laplace operator on semi-discrete surfaces, which are represented by mappings into n-dimensional Euclidean space depending on one discrete and one smooth variable.

Jussi Behrndt:Selfadjoint realizations of the Laplacian on finite and infinite graphs

We define a family of Laplacians on a metric graph and discuss possible
vertex conditions for the functions in the domain of the operators such that the corresponding operator becomes self-adjoint in the L2-space on the graph. For finite graphs and infinite graphs with positive lower bound on the edge length this is well understood; for infinite graphs with edge lengths tending to zero many questions remain open.

Wolfgang Woess:Laplacians on strip complexes

A strip complex is an extension of the concept of metric graphs. It carries the topological structure of the product of a metric graph with a manifold (e.g. ine) and is equipped with additional Riemannian structure in the interior of
the 'strips'. The rigorous construction of Laplacians on such structures is due to Bendikov and Saloof-Coste in collaboration with M. Salvatori and W. Woess. In the talk, the basic concepts along with a key example are presented.

Constellations and hypermaps generalize combinatorial maps, i.e., embeddings of graphs in a surface, in terms of factorizations of permutations. In this talk, we extend a result of Jackson and Visentin (1990) stating an enumerative relation between quadrangulations and bipartite quadrangulations. We show a similar relation between hypermaps and constellations by using a result of Littlewood on factorization of characters. A combinatorial proof of Littlewood's result is also given. Furthermore, we show that the coefficients in our relation are all positive integers, hinting at the possibility of a combinatorial interpretation. Using this enumerative relation, we recover a result on the asymptotic behavior of hypermaps obtained by Chapuy (2009).

Abstract: Consider a system of m forms of degree d in n variables over the integers. A classical result by Birch uses the circle method to provide an asymptotic formula for the number of integer solutions to this system in a homogeneously expanding box, as long as n is large compared to m and d. An analogous result over arbitrary number fields was proved by Skinner. In joint work with M. Madritsch, we extend Skinner's techniques to a recent generalization of Birch's theorem by Browning and Heath-Brown, where they allow the forms to have differing degrees.

We discuss the main ingredients of the proof, as well as consequences of this result to the Hasse principle, weak approximation, and Manin's conjecture.

We discuss two recent methods in which an object with a certain property is sought. In both, using a straightforward random object would succeed with only exponentially small probability. The new randomized algorithms run efficiently and also give new proofs of the existence of the desired object. In both cases there is a potentially broad use of the methodology.

(i) Consider an instance of k-SAT in which each clause overlaps (has a variable in common, regardless of the negation symbol) with at most d others. Lovasz showed for certain d,k (regardless of the number of variables) the conjunction of the clauses was satisfiable. The new approach due to Moser is to start with a random true-false assignment. In a WHILE loop, if any clause is not satisfied we "fix it" by a random reassignment. The analysis (due, basically, to Don Knuth) of the algorithm is unusual, connecting the running of the algorithm with certain Tetris patterns, and leading to some algebraic combinatorics.
[These results apply in a quite general setting with underlying independent ``coin flips'' and bad events (the clause not being satisfied) that depend on only a few of the coin flips.]

(ii) No Outliers. Given n vectors n-space with all coefficients between -1 and +1 one wants a vector x with all coordinates -1 or +1 so that its dot products with all the n vectors are at most K times the square root of n in absolute value, K an absolute constant. A random x would make the dot product Gaussian but there would be outliers. The existence of such an x was first shown by the speaker. The first algorithm was found by Nikhil Bansal. The approach here, due to Lovett and Meka, is to begin with x the all zero vector and let it float in a kind of restricted Brownian Motion until all the coordinates hit
the boundary.

For some of these models, it is not possible in principle to express them as polynomials on Gaussian Matrices and deterministic matrices and hence the usual asymptotic freeness results do not apply in general. However, we observe that as the dimension grows, the models get closer to the model which one would obtain by replacing the independent complex Gaussians by free circulars.

This motivates us to do a more detailed study of the (operator-valued) convergence of complex Gaussian matrices to the circular operator. Motivated by the shape of the matrix models in question, we look first at strictly alternating moments.

The order of the contributions of each partition is given by the number of connected components and the first crossing contribution is given by the partition {{1,4}{2,5}{3,6}}. This differs from the self-adjoint situation where the partition {1,3}{2,4} is the first to contribute. We study some first consequences and generalizations of this phenomenon.

Vortrag für Bachelor- und Masterstudenten

Günter Ziegler described our recent book as ``a
lovely little travel guide to a country you might not even
have heard about -- full of wonders, mysteries, small and
large discoveries.'' We'll sample some gems, including:

1. Stirling's Formula. How do the two great constants $e$ and $\pi$
appear in the asymptotic formula for $n!$. The key: asymptotic
integration.

2. Primes. You know that there are an infinite number of primes.
You probably have not seen the proof given in Asymptopia. Even
more, the sum of the reciprocals of the primes diverges.

Industrial and architectural design processes must respect both performance
requirements and fabrication limitations, but have to offer the maximum
possible creative freedom. In our work we model this situation as
interactive exploration of constraint spaces guided by fairness energies,
and implement it by an iterative Newton method made feasible by conversion
of polynomial constraints to linear or quadratic ones.

We demonstrate different applications to computational design problems which
could not be solved before, such as meshes which obey side-conditions
originating in both geometry and statics. Another instance is developable
surfaces, including curved-crease origami, and sketch-based design.

Strukturtheorie-Seminar

An invitation by examples:
Starting from classical probability, a random
variable can be viewed as an element of a
commutative algebra equipped with a linear
functional to compute the expected value.
Once using $C^*$- and von~Neumann-algebras
as a general framework, one could drop
commutative} and replace the concept of
(classical) independence by free independence}.
Ending up in Free Probability}...

Vortrag im Seminar Diskrete Mathematik und Optimierung

A bootstrap percolation process on a hypergraph $H = H(V;E)$ with activation threshold an integer $r \geq 1$ is a deterministic process which evolves in rounds. Every vertex is in one of two possible states: it is either infected or uninfected.
The set of initially infected vertices is given by $\mathcal{A}(0)$. In each round of the process every uninfected vertex $v$ which has at least $r$ infected neighbours becomes infected. Once a vertex has become infected it remains infected forever. The process stops once no more vertices become infected.

We consider the case when the hypergraph $H$ is a random $k$-uniform hypergraph and $\mathcal{A}(0)$ consists of a given number of vertices chosen uniformly at random.

We generalize results by Janson, {\L}uczak, Turova and Vallier for bootstrap percolation on the binomial random graph. In particular we determine the critical threshold $a_c$ such that if $|\mathcal{A}(0)|\rightarrow (1+\delta)a_c$ then with high probability almost every vertex becomes infected, while if $|\mathcal{A}(0)|\rightarrow (1-\delta)a_c$ then with high probability only a ``few'' additional vertices become infected.

In our talk we will start from the definition of the Riemann zeta function
and recall its basic relationship to the prime numbers. Via the Mellin
transform the Riemann zeta function is related to the classical theta
function, a holomorphic modular of half-integral weight. The theta
inversion formula then provides a tool to meromorphically continue the
Riemann zeta function to the whole complex plane. In the second part
of the talk we will discuss joint work with J. Jorgenson, in which we
establish sup-norm bounds for Maass wave forms, i.e. certain smooth
automorphic forms. These bounds turn out to be related to the
well-known Lindelöf hypothesis.

Seminar TM

\noindent We study polynomial solutions to $T f = \lambda f$, where
$T = \sum_{j=1}^k Q_j D^j$, $D:=d/dz$, and the $Q_j$ are complex
polynomials in a complex variable. We consider two particular cases.

The first is the exactly solvable operator \ie \,when $\deg Q_j \le
j$ with equality for at least one $j$. Every exactly solvable
operator has an eigenpolynomial of any sufficiently large degree. In
the degenerate case, \ie \,if $\deg Q_k < k$, the root of maximal
modulus of the $n$th degree eigenpolynomial $f_n$ tends to infinity
when $n \rightarrow \infty$. It has been conjectured that the root
of maximal modulus $z_n^{max}$ grows as a rational power of $n$, \ie
\,$\lim_{n \rightarrow \infty} z_n^{max} = c_T n^d, c_T > 0$. The
value of $d$ is determined by a particular form of $T$, it depends
on $\deg Q_j, j=1,\ldots,k$. Studying the generic case we are able
to derive explicit expressions for $d$ and $c_T$, formerly only
estimated.

Strukturtheorie Seminar

In this talk I will present results about asymptotic entropy of random walks on regular languages over a finite alphabet. In particular, this setting applies to the case of random walks on virtually free groups. I will discuss the ideas and techniques which are used to prove existence of the asymptotic entropy and which lead to a formula. Moreover, I will discuss how to extend the proof in order to show that the entropy varies analytically in terms of probability measures of constant support. The focus of this talk lies on the discussion of the techniques and the assumptions made at the beginning.

We introduce a new class of holomorphic functions of mixed Hardy-Bergman type on a half-plane. We study some functional properties of these spaces, and obtain some results concerning their zero-sets. We apply these results to the
Müntz-Szasz problem for the Bergman space. We also discuss the question of
describing invariant subspaces, in analogy with the classical theorems of Buerling and Lax in Hardy spaces.

Generalizing pushdown automata, we introduce the notion of multipass automata and study the classes of languages accepted by such machines. The class of languages accepted by deterministic multipass automata is exactly the Boolean closure of the class of deterministic context-free languages while the class of languages accepted by nondeterministic multipass automata is exactly the class of poly-context-free languages, that is, languages which are the intersection of finitely many context-free languages. We illustrate the use of these machines by studying groups whose word problems are accepted by multipass automata. This is a joint work with M.~Coornaert, F.~Fiorenzi and P.E.~Schupp.

\newpage{}

\phantom{x}

\textsl{Graph products and symmetry breaking in graphs}

A coloring of the vertices of a graph $G$ is said to break the symmetries of $G$ if it is only preserved by the trivial automorphism. The minimum number of colors of such a coloring is the distinguishing number of $G$. Despite the fact that the structure of the automorphism group of graph products is well known, there are still open problems about their distinguishing numbers. Many of them pertain to weak products of infinitely many factors. We address such problems for the Cartesian, the strong, and the direct product. On the way we investigate properties of weak products and describe their role in the characterization of median graphs with non-exponential growth and in the game of cops and robbers. Then we turn to countable graphs $G$ with distinguishing 2-colorings where one color class is finite. The minimum number of elements in the finite class of all such colorings is the cost of 2-distinguishing $G$. We characterize such graphs and give bounds on the cost of 2-distinguishing $G$. The best bounds are for two-ended graphs and depend on their product-like properties. This is joint work with Debra Boutin and Ehsan Estaji.

\textsl{Dyson's spike and the spectral measure of groups}

In 1953, Dyson discovered that certain 1-dimensional random Schroedinger operators with gamma potentials have a spike in their spectral measure. We generalize Dyson's results to arbitrary potentials satisfying the central limit theorem. It follows that the switch or walk lamplighter graph on $\mathbb{Z}$, as well as some finitely presented groups, the spectral measure of $(0,x)$ blows up like $1/\log^2(x)$, disproving a conjecture of Lott and Lück. This is joint work with Marcin Kotowski.

The Quasi-Monte Carlo method is a classical method for numerical integration of a function on the multidimensional unit cube. Its basic justification is the
Koksma-Hlawka inequality, which states that the deviation between the integral of a function and the average value of the function, evaluated at a certain set of sampling points, can be estimated in terms of the variation of the function and the so-called discrepancy of the set of sampling points. Consequently, low-discrepancy point sets can be used for numerical integration of ultivariate functions. For several decades, the dimension of the problem was assumed to be fixed, and the cardinality of the point set was allowed to tend to infinity. Recently, special attention has been paid to the case when the dimension is large, and the cardinality of the point set has to remain moderate (in comparison with the dimension). We present some of the known results for this problem together with important proof techniques, which connect the problem with the theory of empirical processes indexed by sets and with metric entropy theory.

The Quasi-Monte Carlo method is a classical method for numerical integration of a function on the multidimensional unit cube. Its basic justification is the
Koksma-Hlawka inequality, which states that the deviation between the integral of a function and the average value of the function, evaluated at a certain set
of sampling points, can be estimated in terms of the variation of the function and the so-called discrepancy of the set of sampling points. Consequently,
low-discrepancy point sets can be used for numerical integration of multivariate functions. For several decades, the dimension of the problem was assumed to be fixed, and the cardinality of the point set was allowed to tend to infinity. Recently, special attention has been paid to the case when the dimension is large, and the cardinality of the point set has to remain moderate (in comparison with the dimension). We present some of the known results for this problem together with important proof techniques, which connect the problem with the theory of empirical processes indexed by sets and with metric entropy theory.

In many applications in numerics and signal processing, data is governed by an\-iso\-tropic structures such as edges in images or shock fronts in hyperbolic PDEs. Due to the complexity of such anisotropic phenomena, their efficient approximation and resolution is challenging and, in particular, standard methods such as finite elements or wavelets fail at this task. In this talk I will present recent constructions from the world of Computational Harmonic Analysis which are capable of approximating curved singularities at an optimal rate. I will illustrate these results with applications in image processing and the numerical solution of transport PDEs.

There exists a canonical symmetric endomorphism $S_p$, the so called shape operator, of the tangent plane of the ellipsoid $\mathbb E$ in $\mafthbb R^3$ at $p\in \mathbb E$ which has as eigenvectors the direction of largest and of smallest curvature on $\mathbb E$. The families of eigenvectors define on $\mathbb E$ two vector fields $X_1$ and $X_2$ which can be integrated and give the so-called curvature lines $K_1$ and $K_2$. They are closed curves on the ellipsoid. For $\alpha\in [0, 2\pi)$ we define a new vector field $X_\alpha= X_1\sin \alpha + X_2\cosin \alpha$ which can be integrated to give a curvature line $K_\alpha$ on the ellipsoid. One can now ask the question for which angles $\alpha$ the curvature line $K_\alpha$ is dense in $\mathbb E$. We give an answer to this question by determining the dimension of the vector space over $\overline{\mathbb Q}$ generated by the periods of elliptic integrals of the first, second and third kind. The latter turns out to solve an extension of a problem of Th. Schneider.

Gastvortrag

We prove optimal bounds for the discretization error of geodesic finite elements for variational partial differential equations for functions that map into a nonlinear space. For this we first generalize the well-known Cea lemma to nonlinear function spaces. In a second step we prove optimal interpolation error estimates for pointwise interpolation by geodesic finite elements of arbitrary order. These two results are both of independent interest. Together they yield optimal a priori error estimates for a large class of manifold-valued variational problems. We measure the discretization error both intrinsically using an $H^1$-type Finsler norm, and with the $H^1$-norm using embeddings of the codomain in a linear space. To measure the regularity of the solution we propose a nonstandard smoothness descriptor for manifold-valued functions, which bounds additional terms not captured by Sobolev norms. As an application we obtain optimal a priori error estimates for discretizations of smooth harmonic maps using geodesic finite elements, yielding the first high order scheme for this problem. {\it (This is joint work with H. Hardering and O. Sander)}

Every finite automaton naturally produces a sequence of finite regular graphs, which correspond to the action of automaton on input words of fixed length. These graphs have very interesting asymptotic and
combinatorial properties, and connect automata to diverse areas of
mathematics. We will discuss some relations which exist between automaton
action graphs and such objects as Gromov hyperbolic spaces, fractal spaces,
zig-zag and replacement graph products, expanding graphs etc.

Abstract: The Titchmarsh-Weyl function for the Laplacian on a metric
graph determines the spectrum of the Laplacian subject to standard
(Kirchhoff) vertex conditions uniquely, provided that the edge lengths
of the graph are rationally independent. We discuss how this property
behaves under small perturbations of the edge lengths.

We single out a special class of infinite graphs with strong compactness type properties. We show that the associated heat semigroup consists of trace class operators and that the associated Dirichlet problem on the Royden boundary of the graph has a unique solution. (Based on joint work with Agelos Georgakopoulos, Matthias Keller, Sebastian Haeseler, Radek Wojciechowski).

Abstract: Only very little is known about the asymptotic
distribution of rational points on smooth cubic surfaces. In
1998, Slater and Swinnerton-Dyer proved lower bounds of
the (conjecturally) correct order of magnitude for rational
points of bounded height on cubic surfaces that contain two skew
rational lines.

We discuss a new straightforward proof of this result that relies
on a fibration of the surface into conics. Our proof has the
additional advantage of working over arbitrary number fields.

In the present talk we want to construct normal numbers with respect
to a given shift invariant measure.

Let $A$ be an (finite or infinite) alphabet and let $T\colon
A^{\mathbb{N}}\to A^{\mathbb{N}}$ be the shift
map defined by $T((a_n)_{n\geq1}=(a_{n+1})_{n\geq1}$. We call
\[[c_1,\ldots,c_k]=\left\{(a_n)_{n\geq1}\in A^{\mathbb{N}}\colon
a_1=c_1,\ldots,a_k=c_k\right\}\] the cylinder set of all infinite
words starting with the letters $c_1,\ldots,c_k$. Given an invariant
probability measure $\mu$ we call $\omega\in A^{\mathbb{N}}$ normal
(generic) with respect to $\mu$ provided that for each cylinder set $C$
\[\frac1N\sum_{n=0}^{N-1}\mathbbm{1}_C(T^n\omega)\xrightarrow[N\to\infty]{}\mu(C),\]
where $\mathbbm{1}_C$ denotes the indicator function of $C$.

The aim of the present talk is to provide a Champernowne type
construction which yields (under some mild conditions) a normal word
with respect to any given shift invariant measure.

Vortrag im Seminar Diskrete Mathematik und Optimierung

We will discuss an algorithm for the single source single sink maximum flow problem in planar graphs with running time $O(n \log n)$ based on the work of Glencora Borradaile and Philipp N. Klein [2009]. The algorithm uses a shortest path tree in the dual planar graph and the corresponding interdigitating tree in the primal graph. Updates in these trees are handled by a dynamic tree data structure to obtain the mentioned running time.

The analysis of the algorithm presented in this talk will be based on a simplified version by Jeff Erickson [2010]. It uses observations of changing crossing numbers during the iterations of the algorithm to bound the number of iterations.

We will also see an overview of some new results by Mozes, Nussbaum and Weimann [2014] which can be used to improve the running time to $O(n \log p)$, where $p$ is the minimum number of edges on any path form the source to the sink.

In this informal talk, I will speak about my job at Wolfram Research, how I got there, and what I am currently doing. I will give a brief introduction to Mathematica and the Wolfram Language, what I think is great about it, and highlight some recent developments. The project I am mostly working on is the Wolfram Cloud, with a web-based version of Mathematica at its core, offering interesting new ways to share documents and deploy software. I will give a quick demo and discuss some of the challenges and lessons learnt. I will also say a few words about working and living in the USA in general.

Abstract:
We consider an economic individual endowed with an initial wealth, having
an income and consuming goods and services. The wealth development rate
is assumed to be a deterministic continuous function of time. The objective
is to maximize the accumulated discounted consumption over a finite time
horizon. The development rate function can take positive as well as negative
values and does not need to obey any monotonic behaviour. This feature
is in contrast to existing results in the mathematical finance and insurance
literature, which generally model the drift rate as a positive function
of the
state variable. The optimization problem can be examined in a purely
analytical
way which allows for a numerical treatment. But, one can do better,
it is possible to derive an algorithm for explicit calculation of the
value function
and optimal strategy by balancing future prospects and immediate
consumption.
It turns out that the value function is in general not continuous.
Finally the
method is illustrated by two examples.

Bootstrap percolation processes were considered for the first time in the late 1970s by Chalupa, Leath and Reich. This is a family of discrete time dissemination processes which evolve in rounds. Given a graph $G$, a natural number $r$ and a subset of vertices of $G$ which we consider as the set of initially infected vertices, in each round a vertex which has at least r infected neighbours becomes infected and remains so for ever. We analyse this process for the case where $G$ is an inhomogeneous random graph (having a rank-1 kernel) and the initial set is uniformly random. We obtain a law of large numbers for the size of the final set. We also consider a special case of the above class of random graphs where the degree sequence follows a power law with exponent between 2 and 3. In this case, we determine a critical function $a_c(n)$ which grows sub-linearly such that if the size of the initial set is $a(n) \gg a_c(n)$, then the final set has linear size, which we determine, but if $a(n) \ll a_c(n)$, then the process does not evolve at all. We also show analogous results for another family of inhomogeneous random graphs, namely random graphs which are created through the preferential attachment model.

Strukturtheorie-Seminar

I will speak about a purely noncommutative criterion for the characterization of two-state normal distribution. I prove that families of two-state normal distribution can be described by relations, which is similar to the conditional expectation in free probability, but has no classical analogue. I also show a generalization of Bozejko, Leinert, and Speicher's formula (relating moments and noncommutative cumulants).

Strukturtheorie-Seminar

The trace of branching random walk (BRW) shares several properties of infinite clusters of invariant percolation processes. In particular, the law of the trace of BRW on free groups (or regular trees) is the law of an infinite cluster of some particular invariant percolation. However, the situation is not as clear for other, especially one-ended, Cayley graphs and many interesting and natural problems remain open.

After a short introduction in the topic we focus on geometric properties of the trace and prove that the trace of a transient branching random walk on a planar hyperbolic group has a.s.~continuum many ends and no isolated end. This property is conjectured (by I. Benjamini) to hold for all nonamenable groups. Our proof is the first for one-ended Cayley graphs and uses crucially the planarity and hyperbolicity of the underlying graph.

Vortrag im Seminar Diskrete Mathematik und Optimierung

In many real-world applications, when modeling network and transportation planning problems, one looks for solutions that should be robust for many different scenarios. For instance, when planning the location of medical service facilities and the corresponding allocation of urban areas to them, decision makers should make decisions before knowing with complete certainty potential location of facilities, or complete demographic characteristics of the surrounding areas. Such kind of uncertainty might arise in dynamic systems subject to geopolitical changes, natural disasters or economical crashes.

This work deals with a facility location problem in which location and allocation policy is defined in two stages such that a first-stage solution should be robust against the possible realizations (scenarios) of the input data that can only be revealed in a second stage. This solution should be robust enough so that it can be recovered} promptly and at low cost in the second stage. In contrast to some related modeling approaches from the literature, this new recoverable robust model is more general in terms of the considered data uncertainty; it can address situations in which uncertainty may be present in any of the following four categories: ``provider-side'' uncertainty, ``receiver-side'' uncertainty, uncertainty ``in-between'', and uncertainty with respect to the cost parameters.

For this novel problem, a sophisticated algorithmic framework based on a Benders decomposition approach is designed and complemented by several non-trivial enhancements, including dual lifting, branching priorities, matheuristics and zero-half cuts. Two large sets of realistic instances that incorporate spatial and demographic information of countries such as Germany and US (in the context of transportation) and Bangladesh and the Philippines (in the context of disaster management) are introduced. They are used to analyze in detail the characteristics of the proposed model and the obtained solutions as well as the effectiveness, behavior and limitations of the designed algorithm.

Joint work with Eduardo Álvarez-Miranda, University of Bologna and Elena Fernández, UPC Barcelona

Strukturtheorie-Seminar

Let $G$ be a group and let $A$ and $B$ be finitely generated subgroups of $G$. In general, the intersection $A\cap B$ need not be finitely generated any more. But, if $G$ is a free group, it is. Here the question arises whether the rank of $A\cap B$ can be bounded in terms of the ranks of $A$ and $B$. In 1956/1957, Hanna Neumann gave such a bound and conjectured that it can be improved by removing the factor 2. This problem became known as the Hanna Neumann Conjecture and it remained unsolved for more than 50 years. Recently, Joel Friedman and Igor Mineyev independently gave proofs of the so called Strengthened Hanna Neumann Conjecture, which implies the classical one.

In this talk, I will give an introduction to free groups and illustrate some important theorems. Then, I will focus on the classical Hanna Neumann Conjecture and give a surprisingly elementary proof developed by Igor Mineyev and Warren Dicks. The talk is designed to address a broad mathematical audience. But, let me underline: It is not about my own research! It is about a brilliant idea that nails an old problem, and deserves to be communicated.

Vortrag im Seminar Diskrete Mathematik und Optimierung

Random planar graphs have attained considerable attention in the recent years. Amongst well studied properties of random planar graphs are connectedness, degree distribution and maximum degree, and the containment of subgraphs.

However, like for the Erd\H{o}s-R\'enyi random graph all the results are for labelled} planar graphs. For unlabelled planar graphs, not even their asymptotic number on a given number of vertices is known. While this number is known for labelled planar graphs, it is not possible to derive the number of unlabelled planar graphs since an unlabelled planar graph can correspond to different numbers of labelled planar graphs due to symmetries of the graph.

Triangulations are the most basic class of planar graphs---by describing their symmetries and the number of triangulations having these kinds of symmetries, we can derive all the information needed to describe symmetries of unlabelled planar graphs and eventually determine their asymptotic number.

It is well known that the real numbers with (purely) periodic decimal expansion are the rationals with denominator coprime to 10. We consider (greedy)
$\beta$-expansions of rational numbers, with $\beta$ being a quadratic Pisot number. Since no rational number has periodic $\beta$-expansion when $\beta$ has a
Galois conjugate in $(0,1)$, we assume that $\beta^2 = a \beta + b$, with $a \ge b \ge 1$. For $b = 1$, Klaus Schmidt (1980) proved that each rational number in
$[0,1)$ has periodic $\beta$-expansion. Shigeki Akiyama introduced the quantity $\gamma(\beta)$, which is the supremum of the numbers $c$ such that each rational
number in the interval $[0,c]$ with denominator coprime to $b$ has periodic $\beta$-expansion. With Milton Minervino, we determined the value of $\gamma(\beta)$
when $a$ and $b$ are coprime, by using a natural extension of the $\beta$-transformation in $\mathbb{R}^2 \times K$, where $K$ is a product of $p$-adic spaces.
Here, we have $\gamma(\beta) = 1$ if and only if $b = 1$. In this talk, we consider the case that $b$ divides $a$. Surprisingly, it is possible here to have
$\gamma(\beta) = 1$ for $b > 1$, namely precisely when $a \ge b^2$ or $b = 6$, $a = 24$ or $a = 30$. For $a < b^2$, the value of \gamma(\beta)$ can be calculated
efficiently with arbitrary precision.
(joint work with Tomáš Hejda)

Strukturtheorie-Seminar

It is an interesting question whether or not one can generate an automaton group which is free. In this talk, I discuss some examples, conjectures, and possible new strategies, developed in collaboration with E. Rodaro (Porto University), to obtain freeness in automata groups.

17.00–17.30 Olaf Steinbach (TU Graz, Austria)
Open problems and challenges from a numerical point of view

Fehmi Cirak (University of Cambridge, United Kingdom)
Multiresolution shape and topology optimisation of shells and solids
The widespread availability of computational engineering software and recent advances in fabrication are enabling the design of optimised structures with ever increasing geometric complexity.
This talk will review our recent work on structural optimisation using multiresolution shape editing and immersed finite element techniques. The geometry of the to be optimised structure is represented with a sequence of increasingly finer surface meshes that can capture geometric details at various resolutions. Using an immersed finite element technique the structure is
analysed without the burden and cost of creating high-quality conforming meshes. Relative to traditional approaches, the fine-grained geometry control provided by multiresolution optimisation results in finding much better performing designs. Moreover, the b-splines employed in multiresolution editing can readily be utilised in computer aided design software for downstream
applications.

Jan Zapletal, Michal Merta (TU Ostrava, Czech Republic)
BEM4I - Parallel BEM Library
In the talk we present a newly developed parallel library based on the boundary element method. The aim of the library is to utilize modern techniques of programming with the main focus on deployment on HPC architectures. To effectively exploit the vector instructions (SSE, AVX) of nowadays CPUs we use explicit vectorization of analytical and numerical evaluation of the boundary integral operators and the representation formula. To speed up the assembly of
Galerkin matrices and to reduce memory consumption we employ the fast multipole method. OpenMP shared memory parallelism is used to assemble both admissible and non-admissible blocks efficiently on multicore CPUs. The distributed memory parallelism by MPI is to be added in the near future. The possible applications of the library include shape optimization problems, where BEM has clear advantages over the finite element method, or sound scattering problems modelled both by the Helmholtz equation and the evolutionary wave equation discretized by the time-domain BEM.

Strukturtheorie-Seminar

Consider an arbitrary homogeneous tree and a subharmonic function on its vertices. We explore the relationship between convex and subharmonic functions on the homogeneous tree. In this talk we investigate several questions concerning the behavior of functions at the boundary and its relation to the behavior of these functions on the set of vertices of the tree.

Mathematisches Kolloquium

Discrete tomography deals with the reconstruction of finite sets from
knowledge about their interaction with certain query sets. The most
prominent example is that of the reconstruction of a finite subset $F$ of
$\mathbb{Z}^d$ from its X-rays (i.e., line sums) in a small positive
integer number $m$ of directions.
Applications of discrete tomography include quality control in
semiconductor industry, plasma particle tracking, image processing,
scheduling, and statistical data security.

The talk will survey some fundamental results in the field and indicate
current lines of research.

Abstract:
For infinitely many primes
$p = 4k + 1$ we give a slightly improved upper bound for the maximal
cardinality of a set B in the cyclic group
$Z_p$ such that the difference set
$B - B$ contains only quadrati cresidues. Namely, instead of the
“trivial” bound
$|B| < \sqrt{p}$ we prove
$|B| <\sqrt{p} - 1$ for approximately three equarters of the primes
$p = 4k + 1$.

Many important NP-hard combinatorial problems can be efficiently
approximated using semidefinite programming relaxations. We propose a new
hierarchy of semidefinite relaxations for classes of such problems that
are based on graphs and for which the projection of the problem onto a subgraph
shares the same structure as the original problem. In particular, we will
focus on the well-studied max-cut problem.

Each level $k$ of the proposed hierarchy consists of the basic semidefinite
relaxation of the problem augmented by the constraints enforcing the
structural projection condition on every $k$-node subgraph of the problem.
This hierarchy has the distinguishing feature that all the relaxations are
formulated in the space of the original semidefinite relaxation.

Preliminary computational results show that the proposed hierarchy yields
improved bounds when compared to the initial relaxation for benchmark
instances of the max-cut and stable-set problems, and that the improved
bounds result in significantly smaller enumeration trees when the relaxation
is used in a branch-and-bound scheme.

Abstract:
We will describe some connections and applications of high rank curves in
families of elliptic curves to several Diophantine problems. This will
include Diophantine m-tuples, i.e. sets of nonzero rationals with the
property that the product of any two of their distinct elements increased
by 1 is a perfect square,
long arithmetic progressions consisting of integers which are solutions of
a Pellian equation, and congruent and theta-congruent numbers.

We study the approximability of the classical quadratic knapsack problem (QKP)
on special graph classes. In this case the quadratic terms of the objective function are not given for each pair of knapsack items. Instead an edge weighted graph $G = (V,E)$ whose vertices represent the knapsack items induces a quadratic profit $p_{ij}$ for the items $i$ and $j$ whenever they are adjacent in $G$ (i.e $(i,j)\in E$).

We show
that the problem permits an FPTAS on graphs of bounded treewidth and a PTAS
on planar graphs and more generally on $H$-minor free graphs. This result is shown by adopting a technique of Demaine et al. (2005). We also show strong NP-hardness
of QKP on graphs that are 3-book embeddable, a natural graph class that is related to planar graphs.

VERSCHIEBUNG ! - Strukturtheorie-Seminar

Bridged graphs are one of the very well investigated graph classes which are applicable also outside graph theory. In this talk different problems connected
with bridged graphs and their generalizations will be presented. I will concentrate on the study of certain complexes, so called bucolic complexes. Their underlying graphs are bucolic graphs, which are a generalization of bridged graphs. I will present some characterizations of bucolic graphs and bucolic complexes and present two properties (contractibility and fixed prism property) of locally finite bucolic complexes.

I will concentrate also on another use of bridged graphs where the relation
between bridged graphs and 3-Steiner convexity will be presented. Within this part I will focus on local convexities and characterize graphs with 3-Steiner convex balls around vertices.

Abstract: We consider the problem of characterizing all number fields
$K$ such that all algebraic integers $\alpha \in K$ can be written as
the sum of distinct units of K. In particular totally complex quartic
fields are of special interest in this talk. One aim of this talk is to
present our methods which come mainly from numeration, beside methods
from algebraic number theory and combinatorics.

Vortrag im Seminar Diskrete Mathematik und Optimierung

Pursuit-evasion based searching, also known as the game of cops and robbers is a game on a graph between two players, $c$ (the cop) and $r$ (the robber). Originally inspired by the problem of searching for a spelunker lost in a cave, the game has a wide range of potential applications from fire fighting to network security.

The rules are as follows: In the first round both $c$ and $r$ choose a starting vertex, in each consecutive round they are allowed to move to a neighboring vertex. The cop wins the game, if after some finite number of steps $c$ and $r$ occupy the same vertex, otherwise the robber wins.

The most basic question related to this game is to characterize the class of cop-win graphs, that is, graphs for which the cop has a winning strategy. For finite graphs this question has been answered by Nowakowski and Winkler who showed that the cop-win graphs are exactly the dismantlable graphs. We present a minor modification to the game due to Chastand et al. which might give a similar characterization for infinite graphs.

Vortrag im Seminar Diskrete Mathematik und Optimierung

This talk is based on an article of Martin Charles Golumbic, Marina Lipshteyn and Michal Stern.

Let $\mathcal{P}$ be a collection of paths on a grid $\mathcal{G}$. Then the edge intersection graph} $\mathit{EPG}(\mathcal{P})$ is a graph, in which the vertices correspond to the paths in $\mathcal{P}$ and there is an edge between two vertices, if the corresponding paths share an edge in the grid $\mathcal{G}$. An undirected graph $G$ is an edge intersection graph of paths on a grid} (EPG), if there exist a grid $\mathcal{G}$ and a collection of paths $\mathcal{P}$ such that $G = \mathit{EPG}(\mathcal{P})$. We call $\langle \mathcal{P},\mathcal{G} \rangle$ an EPG representation} of $G$.

We will see that every graph is EPG. Then we will consider special cases of EPG representations, namely B$_{1}$-EPG representations, in which every path of $\mathcal{P}$ is only allowed to have one single bend, i.e.\ one single turn in the grid. Some of the main results of the paper are that every tree has a B$_{1}$-EPG representation and that there are graphs that do not have a B$_{1}$-EPG representation.

Modern textile production is done using computer controlled machines. The
actions of such machines are subject to mechanical constraints that limit
the possible movements the parts of the machine can execute at a given time.
During the development of a control program for a textile machine, this
limits must be taken into account. For complex patterns it can be non
trivial to find a schedule of movements that resolves the mechanical
constraints and allows the production of the pattern.

In this talk some examples of mechanical problems arising during textile
production will be presented. For some of problems mathematical models will
be presented. In most cases the models lead to NP-complete mathematical
formulations. In some cases the NP-complete problems are solvable for
practical cases due to their relatively small size. Algorithms for solving
practical relevant NP-complete problems will be presented.

Vortrag im Seminar Diskrete Mathematik und Optimierung

A good drawing of a simple graph is a drawing on the sphere or, equivalently, in the plane in which vertices are drawn as distinct points, edges are drawn as Jordan arcs connecting their end vertices, and any pair of edges intersects at most once. In any good drawing, the edges of three pairwise connected vertices form a Jordan curve which we call a triangle. We say that a triangle is empty if one of the two connected components it induces does not contain any of the remaining vertices of the drawing of the graph. We show that the number of empty triangles in any good drawing of the complete graph $K_n$ with $n$ vertices is at least $n$.

A tree parameter is called additive if it can be determined recursively as the sum of the parameter values of all branches, plus a certain toll function. Typical examples include the number of leaves, the number of occurrences of a certain subtree, the total distance of all vertices to the root (the so-called path length), and so on.
We show that such parameters follow a Gaussian limiting distribution for very general additive parameters, provided that the toll functions are bounded and small on average. This applies to different families of trees as well: specifically, simply generated families (binary trees, plane trees, ...), Pólya trees, recursive trees and binary search trees. The results are illustrated by several examples of parameters for which we prove normal or log-normal limit laws.
A typical application is the number of subtrees of a random tree, for which a log-normal law is proven.

The propagation of electromagnetic waves in periodic media is usually
described using Floquet-Bloch theory. This talk will concentrate on a
stack of dielectric slabs which contain a small periodic grating. Such
structures find applications in many integrated optics devices, such
as semi-conductor lasers, waveguide couplers or leaky-wave antennas.

The modes are determined from the spectrum of the Helmholtz operator
on an infinite strip with quasiperiodic boundary conditions, which can
consist of guided modes, radiation modes and leaky modes. To motivate
the periodic case a great deal of attention will be devoted to planar
waveguides which share some of the important features of the periodic
case.

To compute the eigenmodes and the associated propagation constants
numerically, a symmetric boundary integral formulation will be used.
The discretized problem is a nonlinear eigenvalue problem, which is
solved by numerical continuation. The talk concludes with numerical
results for single- and double periodic structures.

Skylines emerged as a useful notion in database queries for
selecting representative groups in multivariate data samples for
further decision making, multi-objective optimization or data
processing, and the k-dominant skylines were naturally introduced
to resolve the abundance of skylines when the dimensionality grows
or when the coordinates are negatively correlated. In this talk
we show that the expected number of k-dominant skylines is
asymptotically zero for large samples when $0<k<d$ under two
reasonable (continuous) probability assumptions of the input points,
d being the (finite) dimensionality, in contrast to the asymptotic
unboundedness when k=d. In addition to such an asymptotic
zero-infinity property, a sharp threshold phenomenon is also
presented for the expected (d-1)-dominant skylines when the
dimensionality is allowed to grow with n. (This talk is based on
joint work with W.-M. Chen and T.-H. Tsai, which is available
at http://dx.doi.org/10.1137/110856952.)

Within numerical optimization, we provide generalized Newton methods for
solution of constrained minimization problems. There is our success to
treat minimization problems which are simultaneously nonconvex and
nonsmooth within the semismooth strategy of primal-dual active sets. In
particular, hemivariational inequalities related to contact with
cohesion phenomena are considered.

In the framework of topology optimization problems we focus on singular
geometric objects. For geometry dependent energy functionals we bridge
from regular perturbations of shape to singular perturbations of
topology by extending the velocity method to nonsmooth velocities. We
refer to topology changes represented by bifurcation phenomena of crack
kinking in fracture.

The problem of identification of unknown geometric objects under
nondestructive testing with acoustic, elastic, electromagnetic waves
belongs to the fields of inverse problems. We endow it with optimization
context. Combining variational techniques and singular perturbations,
for the inverse Helmholtz problem we provide high precision methods of
the object identification.

Optimization methods suitable for the system of nonlinear
Poisson-Nernst-Planck equations are certainly a challenging mathematical
task. We are developing proper variational methods of constrained
minimization, homogenization, and singular perturbation techniques in
nonlinear models related to electrokinetic phenomena in biomolecular,
electrochemical, and photovoltaic systems.